Student's t distribution
The distribution of a sample mean's t-score when σ has to be estimated from the same sample. Like the standard normal it is centred on 0, but its tails are heavier, and they thin out as the degrees of freedom ν grow. Move ν, then ask a question: the area it counts is shaded.
Parameters
Ask a question
Compare with
P(X > 2) = 0.0510
P(X > 2) = 1 − F(2) = 1 − 0.9490 = 0.0510
Summary
| Measure | t(ν = 5) |
|---|---|
| mean | 0.0000 |
| variance | 1.6667 |
| standard deviation | 1.2910 |
| skewness | 0.0000 |
The same in code
- Python (scipy.stats)
t.sf(2, 5)- Excel
=1-T.DIST(2, 5, TRUE)
Formulas
Try this
t* against z*
Open the caffeine preset and turn on the comparison with the normal.
Look for: t* = 2.0096 against 1.96 for the normal. With 49 degrees of freedom the t is already close, but its interval is slightly wider.
Tails as ν grows
Compare with the normal, ask for P(T > 2) and move ν from 1 to 30.
Look for: At ν = 1 the tail beyond 2 is 0.1476, more than six times the normal's 0.0228. By ν = 30 it is 0.0273, close to the normal.
Where the moments go
Set ν = 1, then 2, then 3, and read the summary.
Look for: At ν = 1 the mean is undefined, at ν = 2 the variance is infinite, and the skewness needs ν > 3. Tails this heavy break the usual summaries.
The middle 95 %
With ν = 9, ask for P(−2.262 ≤ T ≤ 2.262).
Look for: The middle area is 0.9500: t* = 2.262 leaves 2.5 % in each tail, which is what a 95 % confidence interval uses.